64 DIFFERENTIAL AND INTEGRAL CALCULUS. 



also /(O) = 1, /' (0) = a ; 



therefore /" (0) = a 2 , /" (0) = a (a" + 1"), 

 and so on, or 



a sin- 1 x iii 2 / ~'*\ / . >.*v x . . B 



..., 

 L- L^ 



which really includes both the series found in the last 

 article, since putting m \/( 1) for a, we get 



a sin- 1 x , . _i N // ^ \ / -i \ 



s =cos(m sm l a?) + J(- 1) Bin(ni sin a 1 ), 



and equating real parts, and also unreal parts, we obtain 

 both expansions. In this case also sin" 1 (x) always denotes 

 an angle between - \ir and TT. 

 Again, since 



a sin- 1 x 



e 



= 1 + a sin" 1 a? + - (sin" 1 a;) 2 + ^ (sin' 1 a;) 3 +. . . , 



L* LI 



we may obtain the series already found for (sin' 1 ^), 

 (sin" 1 xf by picking out the coefficients of , a", and simi- 

 larly we shall get for (sin* 1 xf and (sin" 1 #) 4 , 



... 

 ~ [4 4 U 2 4 



(7 and D being connected in nearly the same way as the 

 series (A), (B) for sin' 1 a- and (sin' 1 a,)*. 



